Literary critic George A. Bonnard actually argues that while, the green world manipulates the city world and makes amends afterword, they actually remain separate and distinct throughout the play.
Bottom is the only character that gets closest to interacting with the fairy world, however, even he awakes believing it was all a dream. Bottom is manipulated by the fairies when Puck turns his head into a donkey's head. Bottom interacts with the fairies when Titania is manipulated through enchantment into falling in love with the first creature she sees, which happens to be donkey-headed Bottom. Bottom interacts with the fairy world through the conversations he has with Titania, through the gifts she bestows on him, and through the conversations he has with the other fairies she orders to be his servants. However, Oberon commands Puck to remove the donkey's head and make Bottom believe that the whole night has been a dream, as we see in Oberon's lines:
[T]ake this transformed scalp
From off the head of this Athenian swain,
That he awaking when the other do
May all to Athens back again repair. (IV.i.63-66)
Hence, we see that while Bottom interacts with the fairies, in the end, because he awakes believing the whole night has been a dream just like the other Athenians, the fairies remain separate and distinct from the human world, including Bottom.
Even though the fairies remain separated from the human world, we do, however, see them making amends for their misdeeds, especially Puck. By the end of the play, the lovers are coupled with whom they belong with. Also, the fairies bless all the lovers who are sharing their wedding nights in Theseus's house. However, since the humans continue to remain completely unaware of the fairies' presence and even existence, we can say that the human world and the fairy world remain separate throughout the play.
The city of Athens, with its patriarchal government, represents the order of the law. In this city, fathers and rulers can tell young women what to do and whom to marry, and the young women simply must obey. In fact, when Hermia wishes to run off with Lysander, she must physically leave Athens, since she could never fulfill her goals there.
The 'green world' on the other hand--the world outside of Athens--is synonymous with wilderness, lawlessness, and an absence of order. In this world, fairies like Puck can play with human emotions. Puck recombines couples multiple times (causing first Lysander and then Demetrius to adore Helena when neither did the day before). Not only do human laws not apply, but natural laws do not even always apply either, since people can be changed so quickly!
At the end of the play, the four lovers must come back to Athens. But what happened in the woods still carries over in some ways: Demetrius still loves Helena, and Hermia and Lysander are still allowed to be together. In this way, the terrifying but liberating lawlessness of the woods meets the strict order of Athens. Theseus, ruler of Athens, relaxes his views and allows goings-on of the woods to become truth in Athens as well.
In many productions, Titania and Oberon (the fairy queen and king of the woods) are played by the same actors as Theseus and Hippolyta (the rulers of Athens). This additional dramatic element also helps to bring them together, since it suggests that maybe the fairy-kingdom isn't actually any different from the world of city government.